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Theorem ressusp 21278
Description: The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Hypotheses
Ref Expression
ressusp.1  |-  B  =  ( Base `  W
)
ressusp.2  |-  J  =  ( TopOpen `  W )
Assertion
Ref Expression
ressusp  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( Ws  A
)  e. UnifSp )

Proof of Theorem ressusp
StepHypRef Expression
1 ressuss 21276 . . . . 5  |-  ( A  e.  J  ->  (UnifSt `  ( Ws  A ) )  =  ( (UnifSt `  W
)t  ( A  X.  A
) ) )
213ad2ant3 1028 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifSt `  ( Ws  A ) )  =  ( (UnifSt `  W
)t  ( A  X.  A
) ) )
3 simp1 1005 . . . . . . 7  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  W  e. UnifSp )
4 ressusp.1 . . . . . . . 8  |-  B  =  ( Base `  W
)
5 eqid 2422 . . . . . . . 8  |-  (UnifSt `  W )  =  (UnifSt `  W )
6 ressusp.2 . . . . . . . 8  |-  J  =  ( TopOpen `  W )
74, 5, 6isusp 21274 . . . . . . 7  |-  ( W  e. UnifSp 
<->  ( (UnifSt `  W
)  e.  (UnifOn `  B )  /\  J  =  (unifTop `  (UnifSt `  W
) ) ) )
83, 7sylib 199 . . . . . 6  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( (UnifSt `  W )  e.  (UnifOn `  B )  /\  J  =  (unifTop `  (UnifSt `  W
) ) ) )
98simpld 460 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifSt `  W
)  e.  (UnifOn `  B ) )
10 simp2 1006 . . . . . . 7  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  W  e.  TopSp
)
114, 6istps 19949 . . . . . . 7  |-  ( W  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
1210, 11sylib 199 . . . . . 6  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  J  e.  (TopOn `  B ) )
13 simp3 1007 . . . . . 6  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  A  e.  J )
14 toponss 19942 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  A  e.  J )  ->  A  C_  B )
1512, 13, 14syl2anc 665 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  A  C_  B
)
16 trust 21242 . . . . 5  |-  ( ( (UnifSt `  W )  e.  (UnifOn `  B )  /\  A  C_  B )  ->  ( (UnifSt `  W )t  ( A  X.  A ) )  e.  (UnifOn `  A )
)
179, 15, 16syl2anc 665 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( (UnifSt `  W )t  ( A  X.  A ) )  e.  (UnifOn `  A )
)
182, 17eqeltrd 2507 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifSt `  ( Ws  A ) )  e.  (UnifOn `  A )
)
19 eqid 2422 . . . . . 6  |-  ( Ws  A )  =  ( Ws  A )
2019, 4ressbas2 15179 . . . . 5  |-  ( A 
C_  B  ->  A  =  ( Base `  ( Ws  A ) ) )
2115, 20syl 17 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  A  =  ( Base `  ( Ws  A
) ) )
2221fveq2d 5885 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifOn `  A
)  =  (UnifOn `  ( Base `  ( Ws  A
) ) ) )
2318, 22eleqtrd 2509 . 2  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifSt `  ( Ws  A ) )  e.  (UnifOn `  ( Base `  ( Ws  A ) ) ) )
248simprd 464 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  J  =  (unifTop `  (UnifSt `  W
) ) )
2513, 24eleqtrd 2509 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  A  e.  (unifTop `  (UnifSt `  W
) ) )
26 restutopopn 21251 . . . 4  |-  ( ( (UnifSt `  W )  e.  (UnifOn `  B )  /\  A  e.  (unifTop `  (UnifSt `  W )
) )  ->  (
(unifTop `  (UnifSt `  W
) )t  A )  =  (unifTop `  ( (UnifSt `  W
)t  ( A  X.  A
) ) ) )
279, 25, 26syl2anc 665 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( (unifTop `  (UnifSt `  W )
)t 
A )  =  (unifTop `  ( (UnifSt `  W
)t  ( A  X.  A
) ) ) )
2824oveq1d 6320 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( Jt  A
)  =  ( (unifTop `  (UnifSt `  W )
)t 
A ) )
292fveq2d 5885 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (unifTop `  (UnifSt `  ( Ws  A ) ) )  =  (unifTop `  (
(UnifSt `  W )t  ( A  X.  A ) ) ) )
3027, 28, 293eqtr4d 2473 . 2  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( Jt  A
)  =  (unifTop `  (UnifSt `  ( Ws  A ) ) ) )
31 eqid 2422 . . 3  |-  ( Base `  ( Ws  A ) )  =  ( Base `  ( Ws  A ) )
32 eqid 2422 . . 3  |-  (UnifSt `  ( Ws  A ) )  =  (UnifSt `  ( Ws  A
) )
3319, 6resstopn 20200 . . 3  |-  ( Jt  A )  =  ( TopOpen `  ( Ws  A ) )
3431, 32, 33isusp 21274 . 2  |-  ( ( Ws  A )  e. UnifSp  <->  ( (UnifSt `  ( Ws  A ) )  e.  (UnifOn `  ( Base `  ( Ws  A ) ) )  /\  ( Jt  A )  =  (unifTop `  (UnifSt `  ( Ws  A ) ) ) ) )
3523, 30, 34sylanbrc 668 1  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( Ws  A
)  e. UnifSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    C_ wss 3436    X. cxp 4851   ` cfv 5601  (class class class)co 6305   Basecbs 15120   ↾s cress 15121   ↾t crest 15318   TopOpenctopn 15319  TopOnctopon 19916   TopSpctps 19917  UnifOncust 21212  unifTopcutop 21243  UnifStcuss 21266  UnifSpcusp 21267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-tset 15208  df-unif 15212  df-rest 15320  df-topn 15321  df-top 19919  df-topon 19921  df-topsp 19922  df-ust 21213  df-utop 21244  df-uss 21269  df-usp 21270
This theorem is referenced by: (None)
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